In the United Kingdom, arms are granted, officially by the Crown, in point of fact by the College of Arms (Scotland having a separate College under the Lord Lyon). While it’s technically incorrect to say that only the heir of line of a given grant of arms may display those arms (his wife, his heir, and various relatives are entitled to use them, properly differenced), neither is it true that anyone descended from an armigerous person may display his arms. You didn’t inherit his castle or estate, so what makes you think you’re entitled to the arms that went with them?
Most European countries, even republics, have preserved a heraldic establishment of some sort to regulate the arms belonging to older and otherwise armigerous families. Ireland established one on becoming independent of the U.K.
The situation in America is quite different. Since sovereignty is vested in the people corporately, and they have not chosen, through Congress or the states, to regulate the use of arms, any citizen may adopt a coat of arms which suits him, often an adaptation of his ancestors’ arms. What he may not do, though this is ethical rather than legal in nature, is to adopt the undifferentiated arms of his ancestors, unless he happens to be the heir of line.
And companies which purport to “discover your family arms for you” are 99% of the time scams, though occasionally you’ll find someone who actually knows heraldry and will give you the straight bill of goods on what your ancestors had, if anything, and what you can properly do with regard to them.
BTW, a “coat of arms” is a garment, a tabard, worked with the particular arms borne by the person wearing it or the person to whom the wearer is in service. It’s considered gauche to use it to refer to any other display of one’s heraldic arms. Properly, the shield bearing the heraldic symbols is called “a shield” or “an escutcheon” – which in noble families and certain other folks specially honored by the Crown is surmounted by a helmet with a crest based loosely on the arms, and separately awarded. Noble families and a few others have supporters – the people or animals at either side of the shield, ostensibly holding it up. If there are supporters, they stand on a ribbonlike strip called the comportment, which may display the family motto. (The motto is something the armigerous person selects for himself, unless the Crown specifically awards one as a mark of honor.) And unlike the other stuff, the motto belongs to all family members. The whole mess taken together is called the achievement, in this case a concrete noun rather than the more common meaning of the word.
Differencing your arms may be done by placing a label – a strip with a (usually odd) number of danglers – across the upper portion of it, by displaying a border, by reversing the colors, or by placing a baton or baton sinister – a diagonal bar crossing the shield but ending before the edge – atop the arms, etc. A white label with three descenders is the mark of the eldest son during his father’s lifetime; one with five descenders marks the eldest son of the eldest son, if he is of age to use arms during his grandfather’s life.
So if you’re descended from a family with argent, a horse rampant gules as the family arms (a red horse rearing on silver or white background), some fourth cousin in England owns the right to use those arms – but as an American, you’re entitled to adapt them and begin using them. So place a wavy red border on the shield, and you have a good-looking set of arms.
As the sole great-great-grandson in my paternal line, which cannot be proven to link back to the rest of the family, but which evidently is connected to an armigerous family that displayed “sable, on a fess argent a boar courant of the first,” I could simply reverse that (which I prefer) to “argent, on a fess sable a boar courant of the first” – but since I’m far from the only descendant of the guy with the arms, I modify it by “…on a fess sable bordered gules…” (This translates to, originally, a black shield, with a white horizontal bar occupying the middle third, and on that bar a black wild boar depicted as running; for me, reverse the black and white colors of the original, and add a thin red border to the black horizontal bar.)